All drugs take effect immediately upon entering the body! A lot of drugs take effect before even entering your bloodstream (usually side effects), e.g. due to irritations they cause on your skin or mucous membranes.
When someone says a drug takes a while to take effect, they are talking about the desirable/therapeutical effects. Just like it will take a while for some undesirable effects to kick in when you are using recreational drugs.
Usually we are talking about adaptive responsive the body undergoes when exposed to the chemical in question or to it's actual primary effects. These responses may then be the therapeutical desired effect itself or they may enable the drug in question to unfold it's desired effects.
These adaptive responses could e.g. be desensitation of a target receptor or it's downregulation, the former of which is the case for SSRI's afaik.
Tbh I hate hearing this crap, even if it is just wrong phrasing ("therapeutical effects" vs. "effects"). part of why I can't hear it anymore is that I get extremely hypomanic minutes after dropping an instant release SSRI (which ironically will wipe out depression). Most people will also get a shitload of side effects on day 1 if they are sensitive enough to the given dosage.
What about an NDRI such as bupropion? I felt the effects about 1-2 hours after my first dose.
The effects of bupropion have been outlined above, but again I'd like to stress that these statements (med x takes time span t to take effect) are not about when you feel a drug, they're about when the desired therapeutical effects kick in. For instance in the case of the lithium I am taking, this can be anywhere between a few days and a year (I think synaptogenesis is supposed to be involved here). Nonetheless most people will feel the drug fuck with your stomach on day 1, oh how they do.
Now there are other substances, e.g. long esters of (lipophilic) steroid hormones, that can take weeks to take effect because they are only released very gradually and even when they are released gradually, their effects are due to profound changes the cell undergoes after their (intracellular) binding to their target proteins, which in turn can mean inhibition or promotion of the synthesis of specific proteins. To finally see the both effects and side effects of e.g. testosterone enanthate manifest will therefore take weeks. Nonetheless side effects can manifest on day one (e.g. an inflammation at the injection site).
This is one example for a substance class directly interfering with "genetic transcription" (which your doctor mentioned). The substance binds to it's target proteins which in turn allows the proteins (if they aren't bound to it already) to bind to certain areas in the DNA. This often promotes or inhibits a process called "transcription", the synthesis of new RNA in this case directly from the respective DNA segment. RNA which is in turn used to synthesize proteins (a process called "translation").